Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some remarks about deformation theory and formality conjecture

Huachen Chen, Laura Pertusi, Xiaolei Zhao

Abstract

Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel--Mukai threefolds and quartic double solids.

Some remarks about deformation theory and formality conjecture

Abstract

Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel--Mukai threefolds and quartic double solids.
Paper Structure (11 sections, 14 theorems, 46 equations)

This paper contains 11 sections, 14 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Deformation theory and good moduli spaces
  3. Preliminaries on deformation theory and DG-Lie algebras
  4. DG-Lie algebra of derived endomorphisms
  5. Kuranishi map
  6. Local structure of a good moduli space
  7. Formality results and moduli spaces
  8. Preliminaries on formality and quasi-cyclic DG-Lie algebras
  9. K3 surfaces
  10. Group actions on categories and Enriques categories
  11. Proof of Theorem \ref{['thm_formality_Enriques_intro']}

Key Result

Theorem 1.1

Let $X$ be a K3 surface and $F\in \mathop{\mathrm{D}^{\mathrm{b}}}\nolimits(X)$ be an object such that $\mathop{\mathrm{Ext}}\nolimits^i(F,F)=0$ for every $i<0$ and the automorphisms group $\emph{Aut}(F)$ is linearly reductive. Then the derived endomorphism Lie algebra $\mathop{\mathrm{RHom}}\nolimi

Theorems & Definitions (31)

  • Theorem 1.1: Theorem \ref{['thm_formality']}
  • Theorem 1.2: Theorem \ref{['thm_formality_Enriques']}
  • Corollary 1.3: Corollary \ref{['cor_qdsandspecGM3']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 21 more